Question

Let A= ????

1 3 2
3 1 2
4 3 2
???? and B = ????
2 1 3
2 2 1
1 3 1
???? represent the bus routes of two companies between three cities.
a. Let C = A+ B. Find matrix C. Explain what the resulting matrix and entry c1,3 mean in this context.
b. Let D = B + A. Find matrix D. Explain what the resulting matrix and entry ????????1,3 mean in this context.
c. What is the relationship between matrices C and D? Why do you think that is?

Answer 1

given,

A =
`\begin{bmatrix}1 & 3 & 2\\3&1& 2\\4 & 3 & 2\end{bmatrix}`

B =
`\begin{bmatrix}2 & 1 & 3\\ 2& 2& 1\\ 1 & 3 & 1\end{bmatrix}`

a) C = A + B

C =
`\begin{bmatrix}1 & 3 & 2\\3&1& 2\\ 4 & 3 & 2\end{bmatrix} +\begin{bmatrix}2 & 1 & 3\\ 2&2& 1\\ 1 & 3 & 1\end{bmatrix}`

C =
`\begin{bmatrix}1 +2 & 3+1 & 2+3\\3+2&1+2& 2+1\\ 4+1 & 3+2 & 2+1\end{bmatrix}`

C =
`\begin{bmatrix}3 & 4 & 5\\5&3& 3\\ 5& 5& 3\end{bmatrix}`

the resulting matrix represent the cities and the route in a single matrix of both the company together.

C₁ ₃ = means city 1 have routes

so, C₁ ₃ = 5

b) D = B + A

D =
`\begin{bmatrix}2 & 1 & 3\\ 2&2& 1\\ 1 & 3 & 1\end{bmatrix} + \begin{bmatrix}1 & 3 & 2\\3&1& 2\\ 4 & 3 & 2\end{bmatrix}`

D =
`\begin{bmatrix}1 +2 & 3+1 & 2+3\\3+2&1+2& 2+1\\ 4+1 & 3+2 & 2+1\end{bmatrix}`

D =
`\begin{bmatrix}3 & 4 & 5\\5&3& 3\\ 5& 5& 3\end{bmatrix}`

D₁ ₃ = means the value of the element on the first row and third column

so, D₁ ₃ = 5

c) both the matrix represent same value

we can say that matrix addition follows commutative law

A + B = B + C